Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.15·2-s − 2.27·3-s − 0.667·4-s + 5-s − 2.63·6-s − 3.20·7-s − 3.07·8-s + 2.19·9-s + 1.15·10-s + 11-s + 1.52·12-s − 5.66·13-s − 3.69·14-s − 2.27·15-s − 2.21·16-s − 4.52·17-s + 2.53·18-s + 4.04·19-s − 0.667·20-s + 7.30·21-s + 1.15·22-s − 6.42·23-s + 7.02·24-s + 25-s − 6.53·26-s + 1.82·27-s + 2.13·28-s + ⋯
L(s)  = 1  + 0.816·2-s − 1.31·3-s − 0.333·4-s + 0.447·5-s − 1.07·6-s − 1.21·7-s − 1.08·8-s + 0.732·9-s + 0.364·10-s + 0.301·11-s + 0.439·12-s − 1.56·13-s − 0.988·14-s − 0.588·15-s − 0.554·16-s − 1.09·17-s + 0.597·18-s + 0.928·19-s − 0.149·20-s + 1.59·21-s + 0.246·22-s − 1.33·23-s + 1.43·24-s + 0.200·25-s − 1.28·26-s + 0.352·27-s + 0.404·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3351463293$
$L(\frac12)$  $\approx$  $0.3351463293$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 - 1.15T + 2T^{2} \)
3 \( 1 + 2.27T + 3T^{2} \)
7 \( 1 + 3.20T + 7T^{2} \)
13 \( 1 + 5.66T + 13T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 - 4.04T + 19T^{2} \)
23 \( 1 + 6.42T + 23T^{2} \)
29 \( 1 - 0.112T + 29T^{2} \)
31 \( 1 + 7.88T + 31T^{2} \)
37 \( 1 + 9.62T + 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 0.751T + 43T^{2} \)
47 \( 1 + 9.75T + 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 - 2.92T + 59T^{2} \)
61 \( 1 - 9.27T + 61T^{2} \)
67 \( 1 + 5.53T + 67T^{2} \)
71 \( 1 - 1.61T + 71T^{2} \)
79 \( 1 - 0.505T + 79T^{2} \)
83 \( 1 - 2.46T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 3.17T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.614881118732483272523535368089, −7.27888755947719129511124737977, −6.65324335319185080589107316751, −6.11965042775613583192202614982, −5.31407054408639648347910192036, −5.00185897157386715947599380368, −3.99726486077451163347121469395, −3.19354488118848048282353940666, −2.10149140530489121823455025970, −0.29321744045911797106052146903, 0.29321744045911797106052146903, 2.10149140530489121823455025970, 3.19354488118848048282353940666, 3.99726486077451163347121469395, 5.00185897157386715947599380368, 5.31407054408639648347910192036, 6.11965042775613583192202614982, 6.65324335319185080589107316751, 7.27888755947719129511124737977, 8.614881118732483272523535368089

Graph of the $Z$-function along the critical line